A picture measures 12cm by 9cm is surrounded by s frame whose area is 100 square centimetres, find the width of the frame.
solve for me this question please
If the width is w, then we know that
(12+2w)(9+2w) - 12*9 = 100
Now just find w.
To find the width of the frame, we need to use the given information that the area of the frame is 100 square centimeters.
Let's assume the width of the frame is "x".
We know that the overall dimensions of the picture and frame together will be increased by twice the width of the frame. Therefore, the overall dimensions will be 12 + 2x and 9 + 2x.
The area of the entire picture and frame can be calculated by multiplying the length and width: (12 + 2x) * (9 + 2x).
We can subtract the area of just the picture (12 * 9) from the area of the entire picture and frame to find the area of the frame: (12 + 2x) * (9 + 2x) - (12 * 9) = 100.
Simplifying the equation, we get (12 + 2x)(9 + 2x) - 108 = 100.
Expanding the equation further, we have (108 + 24x + 18x + 4x^2) - 108 = 100.
Simplifying the equation again, we get 4x^2 + 42x = 100.
Rearranging the equation, we get 4x^2 + 42x - 100 = 0.
Factoring the quadratic equation, we have (2x + 10)(2x - 5) = 0.
Setting each factor equal to zero, we find that 2x + 10 = 0 or 2x - 5 = 0.
Solving these equations, we get x = -5/2 or x = 5/2.
Since the width of the frame cannot be negative, we disregard the negative value.
Therefore, the width of the frame is 5/2 or 2.5 centimeters.
To find the width of the frame, we need to use the given information about the dimensions of the picture and the area of the frame.
Let's assume that the width of the frame is x cm.
We know that the picture measures 12 cm by 9 cm. So, the dimensions of the outer edge of the frame will be (12 + 2x) cm by (9 + 2x) cm since there are two edges on each side.
Now, let's calculate the area of the frame. The area of a rectangle is calculated by multiplying its length and width.
Area of the frame = (12 + 2x) cm * (9 + 2x) cm
But we are given that the area of the frame is 100 square centimetres. So we can set up the equation:
(12 + 2x) cm * (9 + 2x) cm = 100 cm²
Now, we can solve this equation to find the value of x, which represents the width of the frame.
Expanding the equation, we get:
(12 + 2x)(9 + 2x) = 100
Distributing, we have:
108 + 24x + 18x + 4x² = 100
Combining like terms:
4x² + 42x + 8 = 100
Subtracting 100 from both sides:
4x² + 42x - 92 = 0
Now, we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. After finding the possible values of x, we can select the positive value, considering that width cannot be negative.
By solving this quadratic equation, you'll find the width of the frame. The resulting value of x will give you the answer to the problem.